The Class Equation and the Commutativity Degree for Complete Hypergroups

Article The Class Equation and the Commutativity Degree for Complete Hypergroups

Andromeda Cristina Sonea 1,* and Irina Cristea 2,*

1 Faculty of Mathematics, Al. I. Cuza University, Bd. Carol I 13, 700506 Iasi, Romania 2 Centre for Information Technologies and Applied Mathematics, University of Nova Gorica, Vipavska Cesta 13, 5000 Nova Gorica, Slovenia * Correspondence: [email protected] (A.C.S.); [email protected] (I.C.); Tel.: +386-0533-15-395 (I.C.)

Received: 19 November 2020; Accepted: 17 December 2020; Published: 21 December 2020

Abstract: The aim of this paper is to extend, from group theory to hypergroup theory, the class equation and the concept of commutativity degree. Both of them are studied in depth for complete hypergroups because we want to stress the similarities and the differences with respect to group theory, and the representation theorem of complete hypergroups helps us in this direction. We also ﬁnd conditions under which the commutativity degree can be expressed by using the class equation.

Keywords: complete hypergroup; commutativity degree; conjugacy class; class equation

1. Introduction In a non-abelian group—or, more generally, in a non-abelian algebraic structure—it makes sense to compute the probability that two elements commute. This problem was first addressed by Miller [1] in 1944, when he studied the relative number of non-invariant operators in a group. Later on, Erdös and Turan [2] introduced the concept of commutativity degree as the probability that an arbitrary element x in a finite group G commutes with another arbitrary element y in G, that is, d(G) = |{(x, y) ∈ G × G | xy = yx}| /(|G|2). After that, many studies have been developed to determine some bounds for this degree. For example, Gustafson [3] and MacHale [4] proved 5 independently in 1974 that for a non-abelian finite group G, the commutativity degree d(G) ≤ 8 . Other upper bounds for commutativity degree in terms of centralizers have been obtained for the dihedral group Dn by Omer et al. [5]. A classification of the groups for which the commutativity degree is above 11/32 was given in 1979 by Rusin [6], while in 2001, Lescot [7] classified the finite h 1 i groups with d(G) ∈ 2 , 1 , just to recall only some of the studies on this topic. The notion of commutativity degree was generalized in different ways later on. For example, the probability that an element of a given subgroup of a finite group commutes with an element of the group was studied in [8], and the probability that the commutator of a pair of elements of a finite group equals a certain given number was investigated in [9]. The notion of the subgroup commutativity degree of finite groups was proposed by T˘arn˘auceanu[10] as the probability that two subgroups of a given group commute, that is, the probability that the product of two subgroups is again a subgroup. On the other hand, the concept of commutativity degree can also be studied in other algebraic structures, such as in hypergroups. These are a natural generalization of groups, where the operation is substituted by a hyperoperation, i.e., a function defined on the cartesian product of the support set H with values in P ∗ (H), the family of all non-empty subsets of H. Thus, the result of the combination of two elements from the support set is not just an element, as in the classical algebraic structures, but a nonempty subset of the initial set. The hypergroups were introduced by F. Marty in 1934 with their theoretical meaning: The quotient of a group by any of its normal subgroups is again a

Mathematics 2020, 8, 2253; doi:10.3390/math8122253 www.mdpi.com/journal/mathematics Mathematics 2020, 8, 2253 2 of 15 group; while considering a non-normal subgroup, the quotient may be endowed with a hypergroup structure. The recently published paper by Massouros [11] presents the course of development from the hypergroup, as it was initially defined by F. Marty, to the hypergroups endowed with more axioms, and this is used nowadays. The theory of algebraic hypercompositional structures, particularly the hypergroups and hyperrings, becomes then a flourishing area of Modern Algebra, and in the last years, more and more studies in algebraic geometry [12–14], number theory [15], scheme theory [16], tropical geometry [17], and theory of matroids [18] have highlighted the important role of these structures. In this paper, in the context of complete hypergroups, we address two correlated problems: the class equation and the commutativity degree. We consider this particular type of hypergroup because, by using their representation theorem, i.e., Theorem1, we notice that there exists a strong relationship between groups and complete hypergroups. Any complete hypergroup can be obtained using a group, and vice-versa, any group can be viewed as a complete hypergroup. The structure of our study is as follows. First, in Section2, we fix the notation and terminology, and we recall the basic definitions related with the class equation and commutativity degree in group theory, as well as those related to complete hypergroups, which will be used in the following. The class equation for complete hypergroups is established in Section3, while in Section4, we study the commutativity degree for these hypergroups. In addition, we explain that, in order to compute the number of the elements that commute in a finite complete hypergroup, it is necessary to compute the number of the elements that commute in a group, so the commutativity degree of a group influences the commutativity degree of a complete hypergroup (see Theorem4). This connection is also clearly described with several examples that show that the commutativity degree of a complete hypergroup depends on the decomposition of the hypergroup. Thus, the finite complete hypergroups of a certain cardinality and constructed with the same finite group can have different commutativity degrees. Finally, we calculate the commutativity degree using the conjugacy classes (see Theorem5)—the same notion used for determining the class equation. The last section contains the conclusions of our study and some open problems related to it.

2. Preliminaries In this section, we ﬁrst recall the basic notions and results about the class equation and commutativity degree in group theory. Most of them are gathered in the M.A. Thesis of Ref. [19]. Secondly, we brieﬂy present a short review of complete hypergroups, since in the next sections, we will determine the class equation and commutativity degree for such hypergroups. For more details regarding the theory of hypergroups, the reader is refereed to the fundamental books of Refs. [20–22].

2.1. Class Equation and Commutativity Degree for Groups Let (G, ·) be a group and Z(G) = {a ∈ G | ab = ba, ∀b ∈ G} be the center of the group G.

Deﬁnition 1. Ref. [19] We say that two elements a and b are conjugated in G, denoted here by a ∼G b, if there exists g ∈ G such that a = g−1bg.

The relation ∼G is an equivalence relation, and the equivalence class of each element a ∈ G, i.e., [a] = {b ∈ G | ∃g ∈ G : b = gag−1}, is called the conjugacy class of a. For a ﬁnite group G, denote by Sk(G) k(G) the number of distinct conjugacy classes of G, i.e., G = i=1 [xi]. We may recall now the famous class equation in group theory:

k(G) |G| = |Z(G)| + ∑ |[xi]|. (1) i=|Z(G)|+1 Mathematics 2020, 8, 2253 3 of 15

The class equation can be related to another important notion in group theory, one of commutativity degree, which represents the probability that two elements of a group commute [3]. It is deﬁned as follows: |{(a, b) ∈ G2 | a · b = b · a}| d(G) = . (2) |G|2 Let us enumerate some properties of the commutativity degree d(G) [19]:

1.0 < d(G) ≤ 1 for any ﬁnite group G; 2. d(G) = 1 if and only if G is an abelian group; 3. d(G) ≤ d(H)d(G/H) for any ﬁnite group G and H, a normal subgroup of G.

4. The function d(G) is multiplicative, i.e., d(G1 × G2) = d(G1)d(G2), for any two ﬁnite groups G1 and G2. ( ) = k(G) 5. d G |G| .

2.2. Complete Hypergroups Let (H, ◦) be a hypergroupoid, that is, a nonempty set H endowed with a multi-valued operation ◦ : H × H −→ P ∗(H) (also called hyperoperation), where by P ∗(H), we denote the family of nonempty subsets of H. For any nonempty subsets A, B ⊆ H, we denote

[ A ◦ B = (a ◦ b), a∈A,b∈B and thus, if B contains only one element b, we simply write A ◦ b instead of A ◦ {b}. Now, the reproduction axiom and associativity have sense and they appear in the deﬁnition of the following hypercompositional structures:

(i) A semihypergroup is an associative hypergroupoid (H, ◦), i.e., for all a, b, c ∈ H, (a ◦ b) ◦ c = a ◦ (b ◦ c). (ii) A quasihypergroup is a hypergroupoid (H, ◦) that satisﬁes the reproduction axiom: for all a ∈ H, H ◦ a = a ◦ H = H. (iii)A hypergroup is a semihypergroup that is also a quasihypergroup.

In any group, there exists only one identity, and each element has a unique inverse. This property does not hold anymore in hypergroups, in the sense that there may exist (or not) more identities, and each element may have more inverses, or none. An element e ∈ H is called an identity or unit if, for all a ∈ H, a ∈ a ◦ e ∩ e ◦ a. An element a0 ∈ H is called an inverse of a ∈ H if there exists an identity e ∈ H, such that e ∈ a ◦ a0 ∩ a0 ◦ a. A hypergroup that has at least one identity and has the property that, for each element of H, there exists at least one inverse is called regular. The most natural connection between hypergroups and groups is assured by the fundamental relations because the quotient structure of a hypergroup through a fundamental relation is a group with the same properties of the hypergroup. Let us now recall the deﬁnition of one of them, i.e., the β relation: n ∗ n aβb ⇔ ∃n ∈ N , ∃ (x1, x2,..., xn) ∈ H : {a, b} ⊆ ∏xi. i=1 If (H, ◦) is a hypergroup, then β is an equivalence relation, and the quotient H/β is a group. In addition, let ϕH : H −→ H/β be the canonical projection. The heart of a hypergroup H is the set ωH = {x ∈ H | ϕH(x) = 1}, where 1 is the identity of the group H/β. Mathematics 2020, 8, 2253 4 of 15

Let (H, ◦) be a semihypergroup and let A be a nonempty subset of H. We say that A is a complete part of H if the following implication holds:

n n ∗ n ∀n ∈ N , ∀ (x1, ..., xn) ∈ H , ∏xi ∩ A 6= ∅ ⇒ ∏xi ⊆ A. i=1 i=1

Moreover, the complete closure of A in H, denoted by C(A), is the intersection of all complete parts of H that contain A. It can be characterized using the heart of a hypergroup with C(A) = A ◦ ωH = S ωH ◦ A = a∈A β(a). Now, we have all elements for deﬁning the complete hypergroups. Their deﬁnition is based on the concept of a complete part. A hypergroup (H, ◦) is called complete if, for any x, y ∈ H, there is C(x ◦ y) = x ◦ y. Hence, for any x, y ∈ H and a ∈ x ◦ y, we have x ◦ y = β(a). In practice, it is more convenient to use a construction, which is called hereafter called the representation theorem, starting from a given group, as described in the next result.

S Theorem 1. A hypergroup (H, ◦) is complete if and only if H can be partitioned as H = Ag, where G and g∈G the subsets Ag of H satisfy the following conditions:

1. (G, ·) is a group.

2. For all g1 6= g2 ∈ G, there is Ag1 ∩ Ag2 = ∅.

3. If (a, b) ∈ Ag1 × Ag2 , then a ◦ b = Ag1·g2 .

It is worth noticing that, for any group G, there are several non-isomorphic complete hypergroups of the same cardinality, depending on the cardinalities of the subsets Ag with g ∈ G. This property will have a strong inﬂuence on the studies presented in the next sections. For more details and the tables of all non-isomorphic complete hypergroups of order less than 6, see [23,24]. The most important properties of the complete hypergroups are gathered in the following result.

Theorem 2. Let (H, ◦) be a complete hypergroup.

1. The heart ωH is the set of all identities of H, and thus, ωH = Ae, where e is the identity of the group G that appears in the representation theorem of H. 2. H is a reversible and regular hypergroup.

3. The Class Equation for Complete Hypergroups The aim of this section is to establish, similarly to in group theory, the class equation for ﬁnite complete hypergroups based on the notion of conjugation. The starting point is the representation theorem (Theorem1) for complete hypergroups, which assures a strong connection between complete hypergroups and groups. First, we deﬁne the conjugation relation between two elements in an arbitrary hypergroup.

Deﬁnition 2. Let (H, ◦) be a hypergroup. We say that two elements a and b in H are conjugated, denoted here by a ∼H b, if there exists c ∈ H such that

C(c ◦ a) ∩ C(b ◦ c) 6= ∅.

Since in a complete hypergroup (H, ◦), for any two elements a, b ∈ H, we have C(a ◦ b) = a ◦ b, then, based on Theorem1, we may reformulate Deﬁnition2 as follows. Mathematics 2020, 8, 2253 5 of 15

Deﬁnition 3. Let (H, ◦) be a complete hypergroup. We say that the elements a and b are conjugated, denoted here by a ∼H b, if there exists c ∈ H such that

c ◦ a = b ◦ c. (3)

Proposition 1. The relation ∼H is an equivalence relation on the complete hypergroup H.

Proof. It is clear that the relation is reﬂexive, since for a ∈ H, we have a ◦ a = a ◦ a, meaning that a ∼H a. Let us prove now that the relation ∼H is symmetric, and suppose that a ∼H b. It follows that there exists c ∈ H such that c ◦ a = b ◦ c. Since (H, ◦) is a complete hypergroup, there exist (and they are unique) the elements g1, g2, h ∈ G such that a ∈ Ag1 , b ∈ Ag2 and c ∈ Ah. Therefore, it follows that = = Ah·g1 Ag2·h, and based on the second condition of Theorem1, we state that hg1 g2h, meaning that −1 −1 −1 0 g = h g h. Then, h g = g h , implying that A −1 = A −1 . Thereby, there exists c ∈ A −1 1 2 2 1 h ·g2 g1·h h 0 0 such that c ◦ b = a ◦ c . This is equivalent with b ∼H a, so the relation ∼H is symmetric. For proving the transitivity, take a, b, c ∈ H, such that a ∼H b and b ∼H c. Similarly to proving the symmetry, there exist (and they are unique) the elements g1, g2, g3 ∈ G such that a ∈ Ag1 , b ∈ Ag2 and c ∈ Ag3 , and the unique elements h, k ∈ G such that hg1 = g2h and kg2 = g3k, leading to the equalities khg1 = kg2h = g3kh, which mean that a ∼H c, concluding the transitivity. Therefore, ∼H is an equivalence relation.

Based on this proof, a linkage between the conjugated elements in a complete hypergroup and the corresponding conjugated elements in the associated group that appears in the representation of the complete hypergroup can be stated as follows.

Proposition 2. If (H, ◦) is a complete hypergroup obtained from the group G, then a ∼H b if and only if

g1 ∼G g2, where a ∈ Ag1 and b ∈ Ag2 .

Now, we are ready to deﬁne the conjugacy class [a] of an element a of a complete hypergroup (H, ◦) as [a] = {b ∈ H | ∃c ∈ H : c ◦ a = b ◦ c} .

It follows very easily that the number of the distinct conjugacy classes in the complete hypergroup H is the same as the number of the distinct conjugacy classes in the associated group, i.e., k(G) = k(H).

Proposition 3. If (H, ◦) is a ﬁnite complete hypergroup and a ∈ H, then

|[ ]| = | | a ∑ Agi , (4) gi∼G g where a ∈ Ag and g ∈ G.

Proof. In a complete hypergroup H, for any a ∈ H, there exists a unique g ∈ G such that a ∈ Ag. According to Proposition2 and using the deﬁnition of the conjugacy class, we observe that

[ [a] = Agi . (5) gi∼G g

Considering the cardinalities and using the fact that Agi ∩ Agj = ∅, for any gi 6= gj ∈ G, i 6= j, the conclusion immediately follows.

We can state now the main result of this section, namely the class equation for ﬁnite complete hypergroups. Mathematics 2020, 8, 2253 6 of 15

Theorem 3. Let (H, ◦) be a ﬁnite complete hypergroup. The following equation holds:

|H| = |ωH| + ∑ |[a]| . (6) a∈/ωH

Proof. Let G be a ﬁnite group of cardinality n, n ∈ N, n ≥ 2, that appears in the representation theorem of the complete hypergroup H. We have ωH = Ae, where e is the neutral element of the group G. Hence, the complete hypergroup H could be written as     [ [ H = Ae ∪  Ag = ωH ∪  Ag , g6=e g6=e where the union is a disjoint one. S S We intend to check that Ag = [a]. g6=e a∈/ωH S Let x ∈ [a]. Then, there exists a0 ∈/ ωH such that x ∈ [a0]. Hence, there exists g ∈ G, g 6= e a∈/ωH S S S S such that a0 ∈ Ag. Since [a0] = Agi , it follows that x ∈ Ag. So, [a] ⊆ Ag. gi∼G g g6=e a∈/ωH g6=e S S Conversely, from the reﬂexivity of the relation ∼H, it follows that a ∈ [a]. Therefore, a ⊆ [a], a∈H a∈H S S and hence Ag ⊆ [a]. Now, the equality is proved. g6=e a∈/ωH ! S So, H can be written as a disjoint union as H = ωH ∪ [a] . This implies that a∈/ωH

|H| = |ωH| + ∑ |[a]| . a∈/ωH

Notice that, if the group G is abelian, then the conjugacy class of an element from H is the entire set to which it belongs.

Proposition 4. If (H, ◦) is a complete hypergroup and the group that appears in the representation of H is abelian, then [a] = Ag, where a ∈ Ag.

Proof. We easily check this assertion by using the deﬁnition of the conjugacy class [a] = {b ∈ H | ∃ c ∈ H : c ◦ a = b ◦ c}. Thus, there exist, and they are unique, the elements g1, g2, and g3 in G such that a ∈ Ag1 , b ∈ Ag2 , and c ∈ Ag3 . Thereby, Ag3·g1 = Ag2·g3 , which implies that

g3g1 = g2g3 = g3g2 by the commutativity of G, meaning that g1 = g2. So, we conclude that b ∈ Ag1 , i.e., [a] = Ag1 . In the following, we will present an example to highlight the class equation deﬁned above.

3 2 2 2 2 Example 1. Let G =< ρ, σ | ρ = σ = (σρ) = e >= {e, ρ, ρ , σ, ρσ, ρ σ} be the dihedral group D6 of order 6 and let (H, ◦) be a proper complete hypergroup of order 7 constructed with the group G. According to the representation theorem of the complete hypergroups, we may construct more of such non-isomorphic hypergroups depending on the cardinalities of the subsets Ag with g ∈ G that partition the hypergroup. In every case, ﬁve sets Ag will contain one element, while the sixth one will contain two elements (this is the unique possibility to decompose the number 7 as a sum of 6 natural numbers). Moreover, the conjugacy classes of the 2 2 elements of the groups G = D6 are [e] = {e}, [ρ] = {ρ, ρ }, and [σ] = {σ, ρσ, ρ σ}. Let H = {a0, a1, a2, a3, a4, a5, a6}. We analyze the cases: Case I. Set Ae = {a0}, Aρ = {a1}, Aρ2 = {a2, a3}, Aσ = {a4}, Aρσ = {a5}, Aρ2σ = {a6}. Mathematics 2020, 8, 2253 7 of 15

For this representation, according to Equation (5), it results that

ωH = Ae, [a1] = {a1, a2, a3}, [a4] = {a4, a5, a6}.

Therefore, the formula of the class equation gives

|H| = |ωH| + |[a1]| + |[a4]|, equivalently with 7 = 1 + 3 + 3.

Case II. Set now Ae = {a0, a1}, Aρ = {a2}, Aρ2 = {a3}, Aσ = {a4}, Aρσ = {a5}, Aρ2σ = {a6}. According to Equation (5), it follows that

ωH = Ae, [a2] = {a2, a3}, [a4] = {a4, a5, a6}.

Hence, the class equation says |H| = |ωH| + |[a2]| + |[a4]|, meaning that 7 = 2 + 2 + 3. Case III. Take Ae = {a0}, Aρ = {a1}, Aρ2 = {a2}, Aσ = {a3, a4}, Aρσ = {a5}, Aρ2σ = {a6}. Similarly to the previous cases, it follows that

ωH = Ae, [a1] = {a1, a2}, [a3] = {a3, a4, a5, a6}.

Consequently, the class equation gives |H| = |ωH| + |[a1]| + |[a3], or 7 = 1 + 2 + 4. All the other cases are similar to the previous three.

4. Commutativity Degree of Complete Hypergroups The aim of this section is to deﬁne the commutativity degree of a hypergroup and to study it in depth for complete hypergroups because of their strong connection with groups. As already mentioned in the preliminaries, the commutativity degree of a ﬁnite group is the probability that two arbitrary elements of the group commute. Explicitly, if we denote by

c(G) = {(x, y) ∈ G2 | x · y = y · x}, then the commutativity degree can be expressed, as proved in [19], by

|c(G)| d(G) = . (7) |G|2

In a similar way we introduce the commutativity degree for hypergroups. Let (H, ◦) be a ﬁnite hypergroup. Deﬁne the commutativity degree by the formula

{(a, b) ∈ H2 a ◦ b = b ◦ a}| d(H) = . (8) |H|2

Since any complete hypergroup is constructed with use of a group, we intend to study the commutativity degree of complete hypergroups and find a relationship with the commutativity degree of the corresponding group. According to the representation theorem for complete hypergroups, i.e., Theorem1, any finite S complete hypergroup (H, ◦) of cardinality m can be represented as H = Ag, where for any element g∈G g of a finite group (G, ·) of cardinality n, the subsets Ag of H satisfy the relations:

i) For any g1 6= g2 ∈ G, we have Ag1 ∩ Ag2 = ∅;

ii) If (a, b) ∈ Ag1 × Ag2 , then a ◦ b = Ag1·g2 . Mathematics 2020, 8, 2253 8 of 15

Based on this property, the formula of the commutativity degree of a complete hypergroup can be written as n o (a b) | ∃g g ∈ G a ∈ A b ∈ A A = A , i, j , gi , gj , gi gj gj gi d(H) = |H|2 n o (a b) | ∃g g ∈ G a ∈ A b ∈ A g g = g g , i, j , gi , gj , i j j i = . (9) |H|2

So, we immediately notice a relationship between the pair of the elements commuting in the complete hypergroup and the pair of the elements commuting in the group. This suggests us that there may exist a connection between d(H) and d(G) and in the following, which we aim to ﬁnd out. First, we will present a method, which will be described in several steps, for counting the number of the pairs (a, b) ∈ H2 that commute. Let us ﬁx some notations. For an arbitrary set S, we denote its cardinality by |S|. Consider the complete hypergroup (H, ◦) with |H| = m and the group (G, ·) that appears in the representation of H with |G| = n, where m, n ∈ N, m, n ≥ 2. Since we consider only proper hypergroups, we also have the n restriction m > n. In addition, for any i ∈ {1, 2, ... , n}, take |Agi | = mi, and then ∑ mi = m, where i=1 Agi are the subsets that appear in the decomposition of H, as in Theorem1.

The main idea of this method is to emphasize the sets Agi that contain one element, so mi = 1. Then, the remaining sets that contain more than one element, and thus have mi > 1, will be written as a union of another two their subsets: The ﬁrst one contains only one element, and the second one contains all the other elements. By doing this, we can count the number of the elements that commute and belong to a singleton set as |c(G)| = d(G) · |G|2.

Let us start with a general form of the sets Agi , i = 1, 2, . . . , n.

Ag1 = {x1,..., xm1 } = {x1} ∪ {x2,..., xm1 }; = { } = { } ∪ { } Ag2 xm1+1,..., xm2 xm1+1 xm1+2,..., xm2 ; . . = { } = { } ∪ { } Agn xmn−1+1,..., xn xmn−1+1 xmn−1+2,..., xn .

1◦ Step. We count that pairs of elements that commute and belong to sets with only one element. Since we have the equivalences:

◦ = ◦ ⇔ = = − x1 xmi+1 xmi+1 x1 g1gi+1 gi+1g1, for i 1, 2, . . . , n 1 ◦ = ◦ ⇔ = = − xmi+1 xmj+1 xmj+1 xmi+1 gi+1gj+1 gj+1gi+1, i, j 1, . . . , n 1, we count in this step |c(G)| = d(G) · |G|2 (10) pairs of elements that commute. ◦ 2 Step. It is clear that the elements contained in the same set Agi commute between them because their commutativity means gi · gi = gi · gi,, which is always true. Each element of the set Agi commutes with itself, meaning that we count mi elements. Then, the number of different elements that commute between them is equal to m ! A2 = i = (m − ) m mi i 1 i. (mi − 2)! Mathematics 2020, 8, 2253 9 of 15

Therefore, the number of the elements from the set Agi that commute between them and were not ◦ − + − = 2 − counted at the 1 Step is (mi 1) mi mi 1 mi 1. Since we have n different sets Agi , at this step, we count n n 2 2 ∑ mi − 1 = ∑mi − n (11) i=1 i=1 pairs of elements that commute. 3◦ Step. Since all the elements of G commute with the identity e, it follows that a ◦ b = b ◦ a for any a ∈ Ae, b ∈ Ag , and gi 6= e. Now, if we take g1 = e, the number of the ordered pairs of elements from i Ag1 and Ag2 commuting are 2m1m2, but the pairs x1, xm1+1 and xm1+1, x1 were already counted in the ﬁrst step. So, we have only 2m1 · m2 − 2 pairs. Repeating this procedure for all sets Ai, we calculate

n 2 (m2m1 − 1) + 2 (m3m1 − 1) + ... + 2 (mnm1 − 1) = 2∑ (mim1 − 1) (12) i=2 pairs of elements of this step. ◦ 4 Step. It remains to consider the pairs of elements from Agi and Agj with gi gj = gjgi, i 6= j, i, j > 1, which were not considered in the ﬁrst step. It is clear that their number is

not 2 ∑ mimj − 1 = c(i, j). (13) 1

Combining the results of all four cases, we conclude that the commutativity degree of H is

n n 2 |c(G)| + ∑ mi − n + 2 ∑ (mim1 − 1) + c(i, j) = = d(H) = i 1 i 2 . |H|2

After making some simple computations, we ﬁnd

n 2 ∑ mi + 2m1(m − m1) − 3n + 2 + c(i, j) n 2 = d(H) = d(G) + i 1 . m m2 The method that we presented before suggests the following result.

n S Theorem 4. Let H = Agi be a complete hypergroup of cardinality m, where G = {g1, g2, ... , gm} is the i=1 group that appears in the representation of H. For any i = 1, ... , n,, denote |Agi | = mi. Then, the commutativity degree of H has the formula

n 2 ∑ mi + 2m1(m − m1) − 3n + 2 + c(i, j) n 2 = d(H) = d(G) + i 1 , (14) m m2 where c(i, j) = 2 ∑ mimj − 1 , for any i 6= j 6= 1 in {2, . . . , n}. 1

Remark 1. If n = m, i.e., the complete hypergroup H coincides with the group G, we have mi = 1 for any i ∈ {1, . . . , n}, and thus, based on Formula (13), c(i, j) = 0. Substituting this into (14), we get

m 2 m + 2(m − 1) − 3m + 2 + 0 d(H) = d(G) + = d(G), m m2 which conﬁrms the validity of the formula. Mathematics 2020, 8, 2253 10 of 15

In the following example, we will calculate the commutativity degree of some complete hypergroups H, of the same cardinality, m = 7, obtained from the same group G = D6, i.e., the dihedral group of order 6, but with different decompositions. More precisely, the cardinalities of the subsets Ag with g ∈ G that partition the hypergroup are different, as we presented in Example1.

Example 2. Let (H, ◦) be a complete hypergroup of order 7 constructed from the group D6, for which we know 1 that d(G) = 2 [19]. I Case. Consider that we have the following decomposition of H:

Ae = {a0, Aρ = {a1}, Aρ2 = {a2, a3}, Aσ = {a4}, Aρσ = {a5}, Aρ2σ = {a6}. Then, using formula (14), we obtain

1 · 62 + 5 · 12 + 22 + 2 (7 − 1) − 3 · 6 + 2 + 2(2 · 1 − 1) 25 d(H) = 2 = > d(G). 72 49

II Case. Consider that we have the following decomposition of H: Ae = {a0, a1}, Aρ = {a2}, Aρ2 = 31 25 {a3},Aσ = {a4}, Aρσ = {a5}, Aρ2σ = {a6}. Again, using formula (14), we get d(H) = 49 > 49 > d(G). III Case. For the following representation of H, where Ae = {a0}, Aρ = {a1}, Aρ2 = {a2}, Aσ = 23 {a3, a4}, Aρσ = {a5}, Aρ2σ = {a6}, we obtain d(H) = 49 < d(G). Concluding, for this kind of complete hypergroup, we obtain three different values of d(H) and exactly

23 25 31 d(H) = , , . 49 49 49

In the following, we will express the commutativity degree d(H) of a complete hypergroup as a formula involving the conjugacy classes of the elements of H. In group theory, it is known that the ( ) = k(G) ( ) commutativity degree of a ﬁnite group G can be expressed as d G |G| , where k G represents the number of the distinct conjugacy classes of the elements of G. Now, we will explain how and why this formula changes in the class of hypergroups. First of all, we will recall some results concerning groups. For any element g of a group G, we denote by [g] the conjugacy class of g, i.e.

[g] = {h ∈ G| ∃ a ∈ G : h = aga−1}, and by CG(g) the centralizer of the element g in G, i.e., CG(g) = {h ∈ G | gh = hg}, which is a subgroup of G. Then, we have |G| |[g]| = [G : CG(g)] = , |CG(g)| meaning that the product |[g]| · |CG(g)| is constant, or in other words, for any h ∈ [g], it is |CG(h)| = |CG(g)|. This property does not hold anymore for complete hypergroups, as we can notice for the hypergroups described in Example2. For an arbitrary hypergroup H, the centralizer CH (x) of an element x in H is deﬁned as

CH (x) = {y ∈ H | C(x ◦ y) = C(y ◦ x)}, Mathematics 2020, 8, 2253 11 of 15 where in the particular case of the complete hypergroups, the complete closure C(x ◦ y) is equal to x ◦ y. S So, for a complete hypergroup (H, ◦) with the representation H = Ag, for any x ∈ H, there exists a g∈G unique gx ∈ G such that x ∈ Agx and thereby:

CH(x) = {y ∈ H|x ◦ y = y ◦ x} = {y ∈ Agy | Agx·gy = Agy·gx } = S = {y ∈ Agy | gx · gy = gy · gx} = {y ∈ Agy | gy ∈ CG(gx)} = Ag, g∈CG(gx) where we have a disjoint union. It follows that |CH (x)| = ∑ |Ag|. (15) g∈CG(gx)

Theorem 5. If (H, ◦) is a ﬁnite complete hypergroup represented by the group G and |CH(x)| = |CH(y)| for any y ∈ [x], then the commutativity degree of H is

k(H) ∑ |[xi]| · |CH(xi)| = d(H) = i 1 , (16) |H|2

k(H) S where k(H) is the number of the distinct conjugacy classes [xi] of the elements of H, and thus, H = [xi]. i=1

Proof. We know already from the previous section that k(G) = k(H). First, we will determine the k(H) S cardinality of the centralizer c(H) of the complete hypergroup H = [xi]. Therefore, i=1

k(H) 2 2 S c (H) = {(a, b) ∈ H | a ◦ b = b ◦ a}={(a, b) ∈ H | b ∈ CH(a), a ∈ [xi]} i=1 2 S 2 = {(a, b) ∈ H | b ∈ CH(a), a ∈ [x1]} {(a, b) ∈ H | b ∈ CH(a), a ∈ [x2]} 2 ∪ ... ∪ {(a, b) ∈ H | b ∈ CH(a), a ∈ [xk(H)]} k(H) = ∪ Ei, i=1

2 where Ei = {(a, b) ∈ H | b ∈ CH(a), a ∈ [xi]}, for i = {1, ... , k(H)}. Since two distinct conjugacy k(H) classes are disjoint, it follows that the set Ei is disjoint, and therefore, |c(H)| = ∑ |Ei|. i=1 Now let us determine the cardinality of each set Ei. Denote

[ ] = { } ∈ \{ } xi xi1 , xi2 ,..., xip , p N 0 .

= ∈ ( ) |{( ) ∈ 2 | ∈ ( )}| = | ( )| Then, for a xij and b CH a , we obtain xij , b H b CH xij CH xij for 1 ≤ j ≤ p, implying that p |E | = |C (x )| i ∑ H ij . j=1 ∈ [ ] Now, according to the hypothesis, for any xij xi , we have

| ( )| = | ( )| CH xij CH xi , Mathematics 2020, 8, 2253 12 of 15

thereby, |Ei| = |[xi]| · |CH (xi)|, and so

k(H) |c (H)| = ∑ |[xi]| · |CH (xi)| . i=1

Based on the deﬁnition of the commutativity degree d(H), we conclude that

k(H) ∑ |[xi]| · |CH (xi)| = d(H) = i 1 . |H|2

Example 3. Let us continue with Case I of Example2, when [a0] = {a0}, [a1] = {a1, a2, a3} and [a4] = {a4, a5, a6}. We check that the condition |CH (x)| = |CH (y)|, for any y ∈ [x], holds. First, we determine the centralizer of each element of H and we obtain:

CH(a0) = H;

CH (a1) = Ae ∪ Aρ ∪ Aρ2 = {a0, a1, a2, a3} = CH (a2) = CH (a3) ;

CH (a4) = Ae ∪ Aσ = {a0, a4};

CH(a5) = Ae ∪ Aρσ = {a0, a5};

CH(a6) = Ae ∪ Aρ2σ = {a0, a6}.

So, it is clear that the condition in the hypothesis of Theorem5 holds. Now let us calculate the cardinalities of the sets Ei, when

k(H) 3 [ [ c(H) = Ei = Ei, i=1 i=1 with

n 2 o n 2 o E1 = (a, b) ∈ H | b ∈ CH (a) , a ∈ [a0] = (a0, b) ∈ H | b ∈ H ;

n 2 o E2 = (a, b) ∈ H | b ∈ CH (a) , a ∈ [a1] 3 [ 2 = {(ai, b) ∈ H | b ∈ CH (ai)}; i=1 n 2 o E3 = (a, b) ∈ H | b ∈ CH (a) , a ∈ [a4] 6 [ 2 = {(ai, b) ∈ H | b ∈ CH (ai)}. i=4

3 Therefore, we have |E1| = 7, |E2| = ∑ |CH(ai)| = |[a1]| · |CH(a1)| = 3 · 4 = 12, |E3| = i=1 6 ∑ CH(aj) = |[a4]| · |CH(a4)| = 3 · 2 = 6, and then, the commutativity degree is j=4

|E | + |E | + |E | 7 + 12 + 6 25 d(H) = 1 2 3 = = , |H|2 72 49 Mathematics 2020, 8, 2253 13 of 15 which has the same value if we calculate it with the formula (16) in Theorem5:

k(H) ∑ |[ai]| · |CH (ai)| = 1 · 7 + 3 · 4 + 3 · 2 25 d(H) = i 1 = = . |H|2 72 49

In the next example, we will consider the same group G = D6 and a complete hypergroup H of the same cardinality 7 constructed from G, but with a different partition of the sets Ag. Here, the condition expressed in Theorem5 does hold anymore, and the formula (16) is also not true.

Example 4. Consider now the hypergroup in Case III of Example2, where [a0] = {a0}, [a1] = {a1, a2}, and [a3] = {a3, a4, a5, a6}. As in Example3, ﬁrst, we determine the centralizers of the elements:

CH (a0) = H,

CH (a1) = Ae ∪ Aρ ∪ Aρ2 = {a0, a1, a2},

CH (a3) = Ae ∪ Aσ = {a0, a3, a4},

CH (a5) = Ae ∪ Aρσ = {a0, a5},

CH (a6) = Ae ∪ Aρ2σ = {a0, a6}.

We notice that a5 ∈ [a3], but |CH (a5)| 6= |CH (a3)|, so the condition in Theorem5 does not hold. Now, 23 by calculating ∑ |CH (x)|, we obtain that its value is 1 · 7 + 2 · 3 + 2 · 3 + 2 + 2 = 23, so d(H) = 49 , as we x∈H calculated in Example2. If we apply the formula (16), we get

|[a ]| · |C (a )| + |[a ]| · |C (a )| + |[a ]| · |C (a )| 25 d(H) = 0 H 0 1 H 1 3 H 3 = , |H|2 49 which is not the correct value.

5. Conclusions Any ﬁnite group can be partitioned in sets of disjoint conjugacy classes. The class equation states that the cardinality of the group is equal to the sum of the cardinalities of the above-mentioned classes. This result has different applications in group theory; among them, we recall the degree of the commutativity of a group, that is, the probability for two different elements to commute (in a non-commutative group). These two fundamental concepts, i.e., the class equation and commutativity degree, are also related in the hypergroup theory. In this paper, we studied them for complete hypergroups. First, we deﬁned the conjugacy relation on a hypergroup, which is an equivalence relation, proving that, in a complete hypergroup, the number of the distinct conjugacy classes is the same as the number of the distinct conjugacy classes in the group that appears in the construction of the considered complete hypergroup. We then stated the class equation for such hypergroups, which is similar to the class equation for groups, with the difference that the role of the center of the group is played now by the heart of the complete hypergroup. The second part of the paper is dedicated to the study of the commutativity degree of a complete hypergroup. This is the probability that two randomly chosen elements commute in a non-commutative hypergroup. It has been already studied for HX-groups [25] and polygroups [26]. First, we have presented a general method to compute this degree, obtaining a formula that depends on the commutativity degree of the group that appears in the representation theorem of the considered complete hypergroup. It is important to stress the fact that, by using a certain group, more non-isomorphic complete hypergroups of the same cardinality can be constructed. Thus, their commutativity degree is not the same, but if the centralizer of each element in Mathematics 2020, 8, 2253 14 of 15

a conjugacy class is the same, i.e., |CH(y)| = |CH(x)| for any y ∈ [x], then the commutativity degree of the complete hypergroup H can be expressed using the class equation of H (see Theorem5). Moreover, it would be interesting to see under which conditions the commutativity degree of the complete hypergroups of a ﬁxed cardinality and constructed from the same group are maximal, or to ﬁnd a constant c ∈ (0, 1] such that if the commutativity degree is greater than c, then the complete hypergroup is commutative. These problems will be investigated in our future work, since the commutativity aspect is an interesting topic in algebra, and is also studied for polynomials, for example, in [27,28].

Author Contributions: Conceptualization, A.C.S. and I.C.; Funding acquisition, I.C.; Investigation, A.C.S. and I.C.; Methodology, A.C.S. and I.C.; Writing—original draft, A.C.S.; Writing—review and editing, I.C. Both authors have read and agreed to the published version of the manuscript. Funding: The second author acknowledges the financial support from the Slovenian Research Agency (research core funding No. P1-0285). Conflicts of Interest: The authors declare no conflict of interest.

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